Cambridge Past Paper Questions

Find the exact value of ∫(from -π/4 to π/4) 3 cos² 5x dx.

The equation of a curve is xy+y²e^x = 4.

Find the complex numbers z for which (z+4)/(z+4i) is real and |z|= √10. Give your answers in the form z = x+iy, where x and y are real.

Let f(x) = (3a-5x) / ((3a+2x)(2a-x)), where a is a positive constant.

With respect to the origin O, the points A, B and C have position vectors given by OA = i+2j, OB = i+3j-2k and OC = 2i-j+3k. The line l passes thro...

The variables x and y satisfy the differential equation sin 4y dy/dx = x sin 2y sin 3x. It is given that y = π/6 when x = π/2.

The diagram shows the curve y = √x sin 2x for 0 ≤ x ≤ π/2. The curve has a maximum point at M, where x = a. [Figure X.X]

A block of mass 12kg is being pulled by a rope up a rough plane. The plane is inclined at an angle of 20° above the horizontal. The rope pulling th...

Three coplanar forces of magnitudes PN, 5N and 10N act at a point O, as shown in the diagram. The resultant of the three forces has magnitude QN an...

The diagram shows the velocity-time graph of the motion of a cyclist. The graph consists of three straight line segments. The cyclist passes a poin...

A lorry of mass 18000 kg is travelling along a straight road.

When a particle P of mass mkg has speed ums⁻¹, its momentum is 4Ns and its kinetic energy is 16J.

Two particles, P and Q, of masses 0.3 kg and 0.6kg respectively, are attached to the ends of a light inextensible string. The string passes over a ...

A particle X moves along a straight track, starting from a point O at time t = 0. The displacement of X from O at time ts is sm, where s = 3t³/² - 6t.

A crate is being pushed in a straight line along a horizontal surface by a force of magnitude 25 N inclined at 20° above the horizontal. The crate ...

Two particles P and Q, of masses 0.2kg and 0.1kg respectively, are free to move in a straight line on a smooth horizontal plane. P is projected tow...

A particle P of mass mkg is attached to one end of a light inextensible string of length 2.6 m. The other end of the string is attached to a fixed ...

A car is travelling along a straight horizontal road. The car passes through a point A, on the road travelling at a speed of 15ms⁻¹, and then accel...

One end of a light inextensible string is attached to a particle A of mass 3 kg. The other end of the string is attached to a particle B of mass 4k...

A particle P moves in a straight line and passes through the point A at time t = 0. The velocity vms⁻¹ of P at time t seconds is given by v = (2t+1...

A particle P of mass 3 kg is projected with a speed of 8 ms⁻¹ up a line of greatest slope of a rough plane inclined at 30° to the horizontal. P is ...

Two particles P and Q, of masses 0.1kg and 0.3kg respectively, are at rest on a smooth horizontal plane. P is projected directly towards Q with spe...

A van of mass 4500kg is towing a trailer of mass 350kg along a straight horizontal road. The van and trailer are connected by a light rigid tow-bar...

The diagram shows a velocity-time graph which models the motion of a particle. The graph consists of 3 straight line segments. The velocity of the ...

Three blocks P, Q and R, of masses 25kg, 20kg and mkg respectively, are held in equilibrium by three light inextensible strings OP, OQ and OR. The ...

A van of mass 2500kg travelling at speed vms⁻¹ experiences a resistance force of kv² N. The constant power of the van's engine is 62.5 kW.

The diagram shows the vertical cross-section ABC of a rough waterslide. The section AB is a straight line of length 5m inclined at an angle of θ to...

A particle X moves in a straight line. The displacement of X from O at time ts after leaving O is sm, where s = 0.3t² +0.6t for 0≤t≤4.

The masses of the bags of rice made by a company are normally distributed with mean μkg and standard deviation 0.14kg. The probability that the mas...

Last Sunday, teams of runners took part in a charity event. The time taken, in seconds, to run 50m was recorded, correct to 1 decimal place, for ea...

Every Saturday, a particular community holds a ‘Puzzle' event to raise money for a new Leisure Centre. Competitors attempt to solve a puzzle as qui...

In a group of 20 musicians, there are 9 guitarists, 6 pianists and 5 drummers. 6 musicians are selected from these 20 to perform at a concert.

A bag contains 10 marbles, of which 4 are red and 6 are blue. Four marbles are selected from the bag at random, without replacement. The random var...

Rachel has three coins. The first coin is biased so that the probability of obtaining a head when it is thrown is. The second coin is biased so tha...

In Millford, 70% of the residents own a bicycle. A random sample of 160 residents is selected.

A bag contains 4 blue marbles and 12 red marbles. One marble is selected at random from the bag. If this marble is blue, it is replaced in the bag,...

Vehicles approaching a certain road junction from Bromley must go either left, right or straight on. Over time, it is known that 30% turn left, 25%...

The times taken, t minutes, by 300 students to travel to Hollowton College are recorded. The results are summarised in the table below. Time (t min...

Find the number of different ways in which the 10 letters in the word AMALGAMATE can be arranged so that...

Kestrels are birds whose adult wingspans are normally distributed with mean 74.8cm and standard deviation 3.2 cm. A random sample of 120 adult kest...

For a set of 40 values of x, it is found that Σ(x-k) = 836.0, Σ(x−k)² = 25410.8, where k is a constant.

At a large college, all students who study Science also study exactly one of Art or Drama or Music. 20% of these students study Art, 45% study Dram...

A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 3 is obtained. The number of throws taken is denoted by the...

84 people attempt a particular puzzle. The times taken, in minutes, to complete the puzzle are recorded. These times are represented in the cumulat...

Bag A contains 6 red marbles, 5 blue marbles and 1 green marble. Bag B contains 5 red marbles and 3 blue marbles. A marble is chosen at random from...

A company sells bags of pasta. The masses of large bags of pasta are normally distributed with mean 2.50kg and standard deviation 0.12kg. (a) Find ...

A set of friends consists of 7 men and 4 women. Three of the men are brothers: Ali, Ben and Charlie.